While the con­cept of ex­plicit solu­tion can be in­ter­preted mess­ily, as in the quote above, there is a ver­sion of this idea that more closely cuts re­al­ity at the joints, com­putabil­ity. A real num­ber is com­putable iff there is a Tur­ing ma­chine that out­puts the num­ber to any de­sired ac­cu­racy. This cov­ers frac­tions, roots, im­plicit solu­tions, in­te­grals, and, if you be­lieve the Church-Tur­ing the­sis, any­thing else we will be able to come up with. https://​​en.wikipe­dia.org/​​wiki/​​Com­putable_number

Com­putabil­ity does not ex­press the same thing we mean with “ex­plicit”. The vague term “ex­plicit” crys­tal­lizes an im­por­tant con­cept, which is de­pen­dent on so­cial and his­tor­i­cal con­text that I tried to elu­ci­date. It is use­ful to give a name to this con­cept, but you can­not re­ally prove the­o­rems about it (there should be no tech­ni­cal defi­ni­tion of “ex­plicit”).

That be­ing said, com­putabil­ity is of course im­por­tant, but slightly too counter-in­tu­itive in prac­tice. Say, you have two polyno­mial vec­torfields. Are solu­tions (to the differ­en­tial equa­tion) com­putable? Sure. Can you say whether the two solu­tions, at time t=1 and start­ing in the ori­gin, co­in­cide? I think not. Equal­ity of com­putable re­als is not de­cid­able af­ter all (liter­ally the halt­ing prob­lem).

re: differ­en­tial equa­tion solu­tions, you can com­pute if they are within ep­silon of each other for any ep­silon, which I feel is “morally the same” as know­ing if they are equal.

It’s true that the con­cepts are not iden­ti­cal. I feel com­putabil­ity is like the “limit” of the “ex­plicit” con­cept, as a com­mu­nity of math­e­mat­i­ci­ans comes to ac­cept more and more ways of for­mally spec­i­fy­ing a num­ber. The cor­re­spon­dence is still not perfect, as differ­ent fam­i­lies of ex­plicit for­mu­lae will have struc­ture(e.g. alge­braic struc­ture) that gen­eral Tur­ing ma­chines will not.

Polyno­mial-time com­putabil­ity is prob­a­bly closer to the no­tion of ex­plic­it­ness (though still not quite the same, as daoza­ich points out). I don’t know of any num­ber that is con­sid­ered ex­plicit but is not polyno­mial-time com­putable.